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Homoclinic organization in the Hindmarsh-Rose model: A three parameter study.

Roberto Barrio1, Santiago Ibáñez2, Lucía Pérez2

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This study reveals the complex bifurcation atlas of the Hindmarsh-Rose neuron model, proposing a global theoretical description for its bursting dynamics. It uncovers novel homoclinic bifurcation structures and their organizing role in neuronal activity.

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Area of Science:

  • Computational Neuroscience
  • Dynamical Systems Theory
  • Mathematical Biology

Background:

  • Bursting phenomena are characteristic of many fast-slow systems, including neuronal models.
  • The Hindmarsh-Rose neuron model is known to exhibit bursting dynamics involving homoclinic bifurcations, but its global structure remains incompletely understood.

Purpose of the Study:

  • To provide a comprehensive global theoretical description of the bursting dynamics in the Hindmarsh-Rose neuron model.
  • To elucidate the complex homoclinic bifurcation structures within a three-parameter space.

Main Methods:

  • Numerical analysis of the Hindmarsh-Rose model in a three-parameter space.
  • Investigation of codimension-one and codimension-two bifurcation surfaces.
  • Development of a global theoretical scheme based on numerical findings.

Main Results:

  • A complex bifurcation atlas was revealed, extending beyond the fast-slow regime.
  • Surfaces of codimension-one homoclinic bifurcations were found to be exponentially close in the fast-slow regime.
  • Isolas of homoclinic bifurcations and codimension-two bifurcation curves organizing spike-adding processes were identified.

Conclusions:

  • The study proposes a global theoretical description for the Hindmarsh-Rose neuron model's bursting dynamics.
  • Homoclinic bifurcation phenomena, though sometimes distant, fundamentally organize the model's bursting behavior.
  • The identified bifurcation structures provide critical insights into neuronal excitability and dynamics.